Use a graphing utility to graph the polar equation below.Find the area of the region between the loops of r = 6(1 + 2 sin(𝜃))

To solve this problem, we need to follow these steps:

1. Graph the polar equation: First, you need to input the polar equation r = 6(1 + 2 sin(𝜃)) into a graphing utility. This could be a graphing calculator or an online tool like Desmos or GeoGebra. The graph will show a rose curve with 2 loops.

2. Identify the region between the loops: The region between the loops is the area enclosed by the two loops of the rose curve. This is the area we want to find.

3. Find the area of the region: To find the area of the region between the loops, we need to integrate the square of the radius function over the interval that corresponds to the loops. The area A of a polar curve defined by r(𝜃) from 𝜃=a to 𝜃=b is given by the formula A = 1/2 ∫ from a to b [r(𝜃)]² d𝜃.

4. For the given polar equation, the radius function is r(𝜃) = 6(1 + 2 sin(𝜃)). The loops occur when 𝜃 ranges from 0 to π (for the upper loop) and from π to 2π (for the lower loop). So, we need to compute the integral A = 1/2 ∫ from 0 to 2π [r(𝜃)]² d𝜃.

5. Compute the integral: This step requires knowledge of calculus, specifically integration techniques. You can use a calculator or an online tool to compute the integral.

6. Multiply the result by 1/2: The final step is to multiply the result of the integral by 1/2 to get the area of the region between the loops.

Remember, the exact steps might vary depending on the specific graphing utility you are using.